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Dimension prints of fractal sets

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

M. Reyes  and
C. A. Rogers
M. Reyes
Affiliation:
Facultad de Informatica, Universidad Politecnica de Madrid, Campus de Montegancedo, Boadilla del Monte, 28660 Madrid, Spain.
C. A. Rogers
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E6BT
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Abstract

Dimension prints were developed in 1988 to distinguish between different fractal sets in Euclidean spaces having the same Hausdorff dimension but with very different geometric characteristics. In this paper we compute the dimension prints of some fractal sets, including generalized Cantor sets on the unit circle S1 in ℝ2 and the graphs of generalized Lebesgue functions, also in ℝ2. In this second case we show that the dimension print for the graphs of the Lebesgue functions can approach the maximal dimension print of a set of dimension 1. We study the dimension prints of Cartesian products of linear Borel sets and obtain the exact dimension print when each linear set has positive measure in its dimension and the dimension of the Cartesian product is the sum of the dimensions of the factors.

MSC classification

Secondary: 28A78: Hausdorff and packing measures
Type
Research Article
Information
Mathematika , Volume 41 , Issue 1 , June 1994 , pp. 68 - 94
Copyright
Copyright © University College London 1994

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References

1.Besicovitch, A. S. and Moran, P. A. P.. The measure of product and cylinder sets. J. London Math. Soc, 20 (1945), 110–120.Google Scholar
2.Falconer, K. J.. The Geometry of Fractal Sets (Cambridge University Press. 1985).Google Scholar
3.Falconer, K. J.. Fractal Geometry (John Wiley, 1990).Google Scholar
4.Howroyd, J. D.. On dimension and on the existence of sets of finite, positive Hausdorff measure. To appear in J. London Math. Soc.Google Scholar
5.Marstrand, J. M.. The dimension of Cartesian product sets. Proc. Camb. Phil. Soc, 50 (1954), 198–202.CrossRefGoogle Scholar
6.Rogers, C. A.. Hausdorff Measures (Cambridge University Press, 1970).Google Scholar
7.Rogers, C. A.. Dimension Prints. Mathematika, 35 (1988), 1–27.CrossRefGoogle Scholar
8.Taylor, S. J.. On Cartesian product sets. J. London Math. Soc, 27 (1952), 295–304.Google Scholar